The machine is an ex-company laptop, which a colleague rebuilt.
I have no XP discs etc.

When he had done it, he mentioned that the hard disk was making noises, consistent with possible failure.
A crash followed about two months later, BSOD, but I managed to search for the error code and follow advice to get it going (via disk checking/fixing, my memory is vague and I didn't keep the details). This happened again, and a third time.

On the third time, I subsequently found that Internet Explorer would open, and show the first page of the chosen site, but if I clicked in any links in the page it reported that it "has encountered a problem and needs to close".
If I click on the link, the dialog box states:

Other software on the machine still seems to work and, contrary to my expectations, the hard drive hasn't failed now for a few months.

If working properly, the machine would be very useful for my son to use, for homework etc.
My colleague works elsewhere now, so I can't ask him to help.
Given I have no discs, what are the possibilities/options for fixing it?

For example (and I hestitate to say this), would downloading IE7 and copy/pasting iexplore.exe into my setup work? (or cause more problems!)

Try using a different browser like Opera or Firefox and see if they work. If you can surf OK using either of those, then we know the hardware is OK, and that the problem is exclusive to IE. Go to Add/Remove Programs and see if IE7 is listed (it will probably be listed towards the bottom of the list as "Windows Internet Explorer 7"). If it's listed, uninstall it and Windows will revert back to IE6 without losing any favorites or other web settings. If IE6 seems to work OK, then you can either just leave it working, or re-install IE7 by downloading it from this link.
Good luck!